A proximal approach to the Schrödinger bridge problem with incomplete information and application to contamination tracking in water networks
This work addresses contamination estimation in water networks, an important domain-specific problem, but it is incremental as it adapts existing methods to handle partial information.
The authors tackled the discrete Schrödinger bridge problem with partial marginal observations, which lacks strict convexity and prevents direct use of standard methods, by proposing a scalable entropic proximal scheme and developing a theoretical framework; they validated it on contamination tracking in a water network, achieving results on a laboratory-scale setup.
In this work, we study a discrete Schrödinger bridge problem with partial marginal observations. A main difficulty compared to the classical Schrödinger bridge formulation is that our problem is not strictly convex and standard Sinkhorn-type methods cannot be directly applied. To address this issue, we propose a scalable computational method based on an entropic proximal scheme. Furthermore, we develop a framework for this problem that includes duality results, characterization of the optimal solutions, and an observability condition that determines when the optimal solution is unique. We validate the method on the problem of estimating contamination in a water distribution network, where the partial marginals correspond to measured pollutant concentrations at the sensor locations. The experiments were conducted on a laboratory-scale water distribution network.